Let a, b be elements of an abelian group G and let m = |a| and n = |b|. If m and n are relatively prime, then |ab| = mn.
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- 17. Let and be elements of a group. Prove that is abelian if and only if .Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.
- 25. Prove or disprove that every group of order is abelian.13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byExercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.